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Physical Review E

Test of a numerical approach to the quantization of billiards

Barbara Dietz, A. Heine, Vincent Heuveline, Achim Richter

Published Feb 24, 2005• 16 citations
Publisher

Abstract

A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from C(3v) to C3 . Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.

DOI
10.1103/physreve.71.026703

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A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from C(3v) to C3 . Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.
In H. Xu et al. [Phys. Rev. Lett. 110, 064102 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064102], a numerical method is introduced-an extension of the conformal-map method of Robnik [J. Phys. A 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027] for nonrelativistic quantum billiards-for the quantization of relativistic neutrino billiards consisting of a massless noninteracting spin-1/2 particle confined to a two-dimensional domain. We demonstrate in this Letter that this method does not provide solutions of the associated Weyl (Dirac) equation, nor does it fulfill the boundary conditions imposed on the spinor eigenfunctions to ensure confinement of the particle to the domain of the billiard. We review in detail the wave equation, boundary conditions, and quantization of neutrino billiards and derivation of relevant equations to make the proof comprehensible for the general reader. Our results are corroborated with numerical ones for nonrelativistic and relativistic quantum billiards whose shapes depend on a parameter, which allows the study of the properties of their eigenstates as the classical dynamics experiences a transition from regular to chaotic dynamics.
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
In this work, we study a family of fully chaotic billiards that exhibits only rotational symmetries, whose geometry is based on the C3C_3C3​ symmetry system proposed by Leyvraz, Schmit, and Seligman~(LSS) in 1996. Quantum spectral analyses are performed on billiards with symmetry CnC_nCn​~(the billiard repeats itself under rotations of 2π/n2π/n2π/n), where nnn is the symmetry parameter. In these systems, there are subspaces of singlets~(invariant under time reversal) and doublets~(not invariant under time reversal). For the LSS billiard, it has been established both numerically and experimentally that the corresponding subspectra follow the Gaussian Orthogonal Ensemble~(GOE) statistics for singlets and the Gaussian Unitary Ensemble~(GUE) statistics for doublets. From a classical perspective, the shapes of these billiards allow certain subregions of phase space to be visited more frequently by chaotic trajectories, a phenomenon known as stickiness. We investigate the relationship between the fraction of sticky regions in classical phase space and the deviations of the energy subspectra from GOE and GUE statistics. Our results suggest the existence of correlations between the energy distributions associated with different symmetry subspaces. In addition, we discuss aspects related to the superposition of the different energy subspectra.
We present the Quantum Section Method as a quantization technique to compute the eigenvalues and the eigenfunctions of quantum systems. As an instructive example we apply this procedure to quantize the annular billiard. The method uses the symmetry of the system to determine an auxiliary section separating the system into partial regions and computes the Green's functions for Schroedinger's equation, obeying the same boundary conditions imposed on the eigenfunctions of the system. The eigenvalues are obtained as zeroes of a finite real determinant and the eigenfunctions are also determined. The present analytical and numerical results are in total agreement with those obtained by other procedures, which shows the efficiency of the method.